On a conjecture of Wilf about the Frobenius number

Given coprime positive integers $$a_1 < \cdots < a_d$$ , the Frobenius number $$F$$ is the largest integer which is not representable as a non-negative integer combination of the $$a_i$$ . Let $$g$$ denote the number of all non-representable positive integers: Wilf conjectured that $$d \ge \frac{F +1}{F+1-g}$$ . We prove that for every fixed value of $$ \lceil \frac{a_1}{d} \rceil $$ the conjecture holds for all values of $$a_1$$ which are sufficiently large and are not divisible by a finite set of primes. We also propose a generalization in the context of one-dimensional local rings and a question on the equality $$d = \frac{F+1}{F+1-g}$$ .